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### Learning Trajectories in Mathematics

A Foundation for Standards, Curriculum, Assessment, and Instruction

Consortium for Policy Research in Education (CPRE)

- Prepared by
- Phil Daro
- CCSS, member of lead writing team
- Frederic A. Mosher
- CPRE, Sr. Research Consultant
- Tom Corcoran
- CPRE, Co-director
- January 2011

Learning Trajectories

- typical, predictable sequences of thinking that emerge as students develop understanding of an idea
- modal descriptions of the development of student thinking over shorter ranges of specific math topics

Learning Trajectories

- learning progressions which characterize paths children seem to follow as they learn mathematics.
- Piaget’s Genetic Epistemology
- Vygotsky’s Zone of Proximal Development

Development of Learning Trajectories vs. CCSS

- Learning Trajectories begin by defining a starting point based on children’s entering understanding and skills and then working forward
- CCSS were begin at the level of college and career ready standards backwards down through the grades. This mapping is based on a logical rendering of the set of desired outcomes needed to define pathways or benchmarks to the standard.

Learning Trajectories

- Are too complex and too conditional to serve as standards. Still learning trajectories point to the way to optimal learning sequences and warn against the hazards that could lead to sequence errors.

Based on Doug Clements’ & Julie Sarama’s in Engaging Young Children In Mathematics (2004).

Pre-Composer

- Free exploration with shapes
- Manipulation of shapes as individuals
- No combining of shapes to compose larger shapes

Picture Composer

- Matches shapes
- Puts several shapes together to make one part of a picture
- Uses “pick and discard” strategy, rather than intentional action
- Notices some aspects of sides but not angles.

Picture Maker

- Moves from using “pick and discard” strategy to placing shapes intentionally.
- Good alignment of sides and improving alignment of angles

Shape Composer

- Combines shapes to make new shapes with anticipation.
- Chooses shapes using angles as well as side length.
- Intentionality based.

Substitution Composer

- Creates different ways to fill a frame emphasizing substitution relationships.

Learning Trajectory for Composing Geometric Shapes

- Pre-composer: Free exploration with shapes
- Picture maker: Makes one part of a picture (arms on pattern block person but not legs)

3. Shape composer: More advanced. Chooses shapes with certain angles and length of sides. “I know that will fit!”

4. Substitution composer: yet more advanced. Can take hexagon outline and fill it in different ways to make a hexagon with pattern blocks.

Trajectories can be used

to develop instructional tasks that:

- support student movement of understanding from one level to another in specific ways
- elicit and assess student understandings

The blank puzzle illustrates the type of structure that will challenge and help a child move their skills along the trajectory

Some Trajectories

- Present a continuum of tasks that are well connected and build on each other in specific ways over time
- Present tasks that connect across topical areas of school math
- Offer detailed guidance to teachers in understanding the capacities and misconceptions of their students at different points in their learning of a particular topic.

Aim of Trajectories

- Are chronologically predictive
- In the sense of what students do (or are able to with appropriate instruction) move successfully from one level to the next
- Yield positive results
- for example deepened conceptual understanding and transferability of knowledge and skills as determined by assessment
- Have learning goals that are mathematically valuable
- align with broad agreement on what math students ought to learn (as reflected in the CCSS)

Trajectories Might Serve CCSS

- by defining more clearly the agreed upon goals for which specific learning trajectories must still be developed because they describe pivotal concepts of school math

Getting the sequence right is not guaranteed

- It involves testing hypothesized dependency of one idea on another, with particular attention to areas where cognitive dependencies are potentially different from logical dependencies as a mathematician sees them

Learning Trajectory Researchers

- Are answering questions about when instruction should follow a logical sequence of deduction from precise definitions and when instruction that builds on a more complex mixture of cognitive factors and prior knowledge is more effective

Value of Learning Trajectories

- Offer a basis for identifying interim goals that students should meet
- Provide understandable points of reference for designing assessments that point to where students are, rather than merely their final score.
- Adaptive instruction thinking your sole goal is to gather actionable information to inform instruction and student learning, not to grade or evaluate achievement
- Could help teachers manage a wide variety of individual learning paths by identifying a more limited range of specific types of reasoning for a given type of problem.

Number Core Trajectory

- Seeing how many objects there are (cardinality)
- Knowing the number word list (one, two, …)
- 1-1 correspondences when counting
- Written number symbols

Multiplicative Reasoning and Rational Number Reasoning

- Equi-partitioning
- Multiplication and division
- Fraction as number
- Ratio and Rate
- Similarity and Scaling
- Linear and Area measurement
- Decimals and Percents

Multiplication Strategies

- Count all
- Additive calculation
- Count by
- Patterned based
- Learned products
- Hybrids of these strategies

Spatial Thinking

- In, on, under, up and down
- Beside and between
- In front of, behind
- Left, right

Measurement

- Compare sizes
- Connect number to length
- Measurement relating to length
- Measuring and understanding units
- Length-unit iteration
- Correct alignment with ruler
- Concept of the zero point

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